Initial, Secondary, and Final Target Interactions

There are an infinitely large number of reflections and transmissions that proliferate in a GPR scene when considering energy incident upon a target surface.

Realistically, all of these interactions can not be accounted for. Even using ray-tracing, the amplitude associated with each propagated ray tends to be significantly small and undetectable by the receiver array. As mentioned earlier, a structured plan of evaluation must be defined in order to account for those rays which maintain enough power to be detected, their phase recorded and collected for pre-image processing. This plan is outlined as a definition of Initial Target Interactions (ITI), Secondary Target Interactions (STI), and Final Target Interactions (FTI). At each of these points, both specular and diffuse energy interactions are considered with respect to the transmitting antenna and receiver array. The BRDF and the geometric analysis of reflected rays inside and outside of the modeled target are discussed in detail. This plan accounts for the accuracy in return power recorded at the receiver array.

The verity of the target model relies heavily on calculating the BRDF at a number of locations. These locations are defined as the center of each triangular facet used to describe the target in 3D space. Both the size of the facet area and distance between each center point is dictated by the SAR resolution. Consequently, this resolution is a function of the physical receiver antenna size (3-4). Based on the BRDF geometry, the angle definitions and the Fresnel Spectral approximation made by Kapfer, the BRDF is calculated at each target boundary using the composite relationship in Equation (5-4).

( 2 2 )2 2 2 2 2 ( ||)

R ≡ Perpendicular polarization power term

R ≡ Parallel polarization power term α ≡ Angle from microfacet normal to permittivity, the specular and directional controls, and the three critical locations (source, incident plane, and observer). For the GPR scene, these parameters fit well into the simulation model since only the receiver (or observed) location is varied. To consider a point source target, a 3D color representation is helpful in visualizing the return power distribution of the BRDF.

Figure 5.9 Three dimensional BRDF representation (Image reproduced from [11] with authors permission)

This point source exhibits isotropic reflections for a given transmitter and receiver array and shows the uniform distribution of return power across the receiver grid. A majority of the energy is located at the center of the synthetic receiver indicating specular reflections, while the diminishing intensities are representative of more diffuse reflections. The locations of these intensities make use of the roughness and isotropy control variables discussed in detail by Kapfer [11].

In determining ITI locations, information for each triangular facet of the target is extracted and stored based on the transmitter and target location. The uniqueness of the geometry requires knowing these fundamental descriptors:

1. The [x y z] coordinates for each of the three points that describe the facet.

2. The center point of the facet.

3. The normal vector describing the direction of the facet in space.

4. The D coefficient to describe the distance of the facet from the origin.

5. Flag values used to indicate whether incident energy will totally internally reflect or partially refract through the facet.

Similar to work by Jeter [1], each facet is represented mathematically by a 6 x 3 grid of data as shown in Table 5.1.

Table 5.1 Facet information storage format

Previous work was limited in modeling targets with the basic shape of a cube, only providing for variations in height, length, or width. A significant upgrade to the GPR simulator is the capability to model targets of any size and shape. This is accomplished using Solid Works, and provides the end-user the ability to create an expanded variety of shapes, assuming the individual facet size corresponds to the system resolution. With this extended capability comes an inherent complexity in the individual analysis of each facet with respect to the transmitter and receiver array, thus the need for each of the columns of data outlined in Table 5.1. This target information is used throughout analysis of the remaining target interactions however it is deliberately collected during the ITI phase of simulation. This allows the information to be segregated into those facets which will totally internally reflect incident power and those that will not.

Using an iterative process, the BRDF is calculated at each receiver for an individual target facet. Taking into consideration the attenuation experienced as a ray propagates through the ground medium, the amount of power a particular receiver will observe is determined. Since each receiver has a different position in the array, this

BRDF value is subject to variation, especially depending on the orientation of the target facet. This is seen in Figure 5.10.

Ray record

ed

Figure 5.10 Ray-Target facet intersection: Valid versus Invalid

Here, the amount of power observed from a facet directed towards the receiver array will be significantly greater than that from a facet angled horizontally and away from the receiver array. Whether or not total internal reflection occurs, the aim of ITI analysis only requires that the observed power by each receiver be recorded for every facet that is guaranteed to reflect energy. The GPR simulator accounts for those facets which are directed at an angle that would prevent any energy from being observed by any receiver in the array. These interactions are ignored since they do not contribute to the total return.

Once a facet has been identified as an ITI which does not totally internally reflect, it is inferred that a percentage of incident power will refract through that facet in the form of a ray. STI and FTI analysis immediately follow this in a cascaded sequence of “if”

statements. Before any additional return energy is calculated, it must be determined whether the refracted ray will propagate back through the GPR scene and intersect with a receiver. Again, there are a potentially large number of rays which may propagate

through the modeled scene; however, these rays are of no interest unless they can be traced back to the receiver array.

Both STI and FTI calculations rely heavily on previous work by Glassner. The intersection between the refracted ray and another target facet (from within the whole target) is calculated and the intersection point is recorded. Glassner’s method for accomplishing this is perfectly suited for this application as it requires determining the intersection between a straight line and a triangular plane. Jeter explains this method as it relates to this application [1]. While only one STI will be identified for each ray refracted through the target initially, the complexities associated with the BRDF become a factor in identifying valid FTI facets.

The occurrence of an STI indicates that energy is reflected in both specular and diffuse forms from inside the target as shown in Figure 5.11.

Figure 5.11 STI Occurrence

As a result, during FTI analysis, two questions must be answered in the affirmative to consider recording any power at the receiver array beyond ITI reflections: 1) Do any of the surrounding facets observe reflected energy internally from the STI? If so, which of these will refract a portion of the observed energy to the receiver array? These are identified as valid FTI facets. The receiver array is segmented into individual triangular

facets and once again, Glassner’s techniques are implemented to determine the intersection between ray refracted from the FTI and the appropriate receiver in the grid.

This is depicted in Figure 5.12.

To rece

iver

Figure 5.12 FTI facet refraction towards receiver

Throughout STI and FTI analysis, the attenuation of the ray as it propagates through the target and back to the receiver array is recorded. Both a power value associated with the attenuated ray and a phase value associated with the path of propagation are stored as a received signal by the simulator. Combined with the reflections from the ITI facets, this information is sent as a set of modulated chirp signals to the pre-image processor.

This format of analysis and sequence of evaluation provides a more geometrically accurate target model and furthers the development of a valid GPR simulation tool.

Chapter 6

Analysis and Results

The uniqueness of this target model as compared to previous models led to complications in pre-image processing and SAR processing, thus a more direct approach was taken to analyze the return data. While many of the principles related to SAR are inherent in the results displayed in Chapter 6, the pre-image processing and SAR processing methodology are not implemented. Instead, target modeling is accomplished by interpolating the known raw return power from the target and true target facet positions to produce visual estimate of the energy returned.